Stabilized oscillator circuit for plasma density measurement

ABSTRACT

A method and system for controlling electron densities in a plasma processing system. By applying a dither voltage and a correction voltage to a voltage-controlled oscillator, electron (plasma) density of a plasma processing system (acting as an open resonator) may be measured and controlled as part of a plasma-based process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.10/031,373, filed Apr. 25, 2002, now U.S. Pat. No. 6,646,386, which is a371 of International Application Serial No. PCT/US00/19540, filed Jul.20, 2000, which claims the benefit of U.S. Provisional Application Ser.No. 60/166,418, filed Nov. 19, 1999. The present application is relatedto U.S. provisional application Ser. No. 60/144,880, filed Jul. 20,1999, U.S. provisional application Ser. No. 60/144,878, filed Jul. 21,1999, and U.S. provisional application Ser. No. 60/144,833, filed Jul.20, 1999. All of those applications are incorporated herein by referencein their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention provides a method and system for measuring andcontrolling electron densities in a plasma processing system, such as isused in semiconductor processing systems.

2. Description of the Background

Following the Second World War, several university research groups usedmicrowave technology that had been developed during the war to studypartially ionized gases. In particular, Professor Sanborn C. Brown'sgroup at Massachusetts Institute of Technology developed and exploitedthe so-called “cavity technique” for the measurement of electron densityin partially ionized, electrically quasi-neutral gases, which have cometo be called plasmas.

In this procedure, changes in the resonant behavior of a microwavecavity were studied as a consequence of the presence of a plasma withinit. Typically, a right, circularly cylindrical cavity operating in itslowest or nearly lowest order resonant mode was used, and the gas wascontained within a coaxial Pyrex™ or quartz tube. An aperture wasprovided in each planar end surface to permit passage of the tubethrough the cavity.

The presence of a plasma within a microwave cavity will, in general,affect both the resonant frequency of a particular cavity mode and thesharpness (Q) of the resonance; i.e., the precision with which thefrequency of a microwave signal must be fixed if the resonant mode is tobe appreciably excited. Using a form of perturbation theory, it ispossible to relate the changes in these parameters to the electrondensity and the electron collision frequency in the plasma. Theperturbation theory is valid only for (radian) frequencies that satisfythe condition:

ω²>>ω_(p) ²≅3.18×10⁹ N _(e)

where ω_(p) is the plasma (radian) frequency, and N_(e) is the electrondensity in electrons/cm³. Consequently, for the diagnosis of plasmaswith electron densities of the order of 10¹² cm⁻³, the magnitudes ofinterest here, a microwave signal frequency (ω/2π) in excess of tens ofGHz is required.

The requirement of signal frequencies on the order of tens of GHz causesa significant problem. The physical dimensions of a cavity designed toresonate in its lowest or nearly lowest order resonant mode are on theorder of the wavelength of the signal. Thus, a cavity designed toresonate at about 35 GHz has dimensions on the order of only acentimeter. The use of such a small cavity for electron densitymeasurements is difficult.

In principle it is possible to use a cavity designed to resonate in a“higher order” mode to overcome the problem associated with the smallphysical size of a lowest or low order mode. However, if this approachis taken, it becomes extremely difficult to know with certainty theidentity of a particular excited cavity mode. Consequently, it becomespractically very difficult, if not impossible, to apply perturbationtheory to determine the electron density and the electron collisionfrequency.

One way to circumvent this problem is to use an “open” resonator, i.e.,a resonator in which the electromagnetic field is not confined by a(nearly) completely enclosing conducting surface. A practical example ofan open resonator is a pair of large aperture, circularly symmetricalend mirrors, with planar or curved surfaces and with no confiningcircularly symmetrical conducting surface between them. Open resonatorsof this type were considered in great detail by A. G. Fox and T. Li in“Resonant modes in a MASER interferometer,” Bell System TechnicalJournal, vol. 40, pp. 453-488, March 1961. Fox et al. showed that anymode that could be regarded as including a plane wave componentpropagating at a significant angle with respect to the axis of symmetrywould not be appreciably excited, i.e., would have a very low Q. Ineffect, for an open resonator, the number of practically useful modeswith resonant frequencies in a particular frequency range is far lessthan the equivalent number for a closed resonator of similar size. Thisproperty of open resonators provided an enormous opportunity forresearchers to extend resonant plasma diagnostic techniques tofrequencies above 35 GHz.

Microwave energy may be coupled from a waveguide feed to an openresonator using the same principles that govern coupling from awaveguide feed to a closed resonator. The location, spatial rotation,and size of a coupling aperture in a resonator mirror has to beappropriately related to the configuration of the electromagnetic fieldfor the desired resonator mode. The input and output coupling aperturesmay both be on the same mirror or the input aperture may be on onemirror and the output aperture on the other.

Known electronically tunable microwave oscillators are frequencystabilized with the aid of a resonant cavity and a microwavediscriminator. The basic concepts are documented in detail in variousM.I.T. Radiation Laboratory Reports and in the Radiation LaboratorySeries published by McGraw-Hill in 1947. One use of those oscillators isto cause an electronically tunable oscillator to track the resonantfrequency of a microwave resonator as that frequency is changed. Anextensive discussion of the techniques is presented in Vol.11, Techniqueof Microwave Measurements, M.I.T. Radiation Laboratory Series, Carol H.Montgomery, Editor, McGraw-Hill Book Company, New York, 1947, pp.58-78(hereinafter “Montgomery”). The entire contents of Montgomery are herebyincorporated by reference. The use of a microwave interferometer is alsodescribed in two known publications: (1) “A Microwave Interferometer forDensity Measurement Stabilization in Process Plasmas,” by Pearson etal., Materials Research Society Symposium Proceedings, Vol. 117 (Eds.Hays et al.), 1988, pgs. 311-317, and (2) “1-millimeter waveinterferometer for the measurement of line integral electron density onTFTR,” by Efthimion et al., Rev. Sci. Instrum. 56 (5), May 1985, pgs.908-910.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an improved plasmaelectron density measurement and control system using a microwaveoscillator locked to an open resonator containing a plasma.

These and other objects of the present invention are achieved by acircuit for stabilizing the oscillation frequency of avoltage-controlled oscillator (VCO) as part of a system for measuringelectron (plasma) density.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will become readily apparent with reference to thefollowing detailed description, particularly when considered inconjunction with the accompanying drawings, in which:

FIG. 1 is a graph showing normalized transmission;

FIG. 2 is a graph of a discriminator output;

FIG. 3 is block diagram of a dither control circuit according to a firstembodiment;

FIG. 4A is a VCO bias signal processor;

FIG. 4B is a timing diagram showing control signals according to thepresent invention;

FIG. 5 is a block diagram showing an expanded system diagram accordingto a second embodiment; and

FIGS. 6A and 6B are block diagrams of alternate embodiments ofself-clocking VCO bias signal processors.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to the drawings, in which like reference numeralsdesignate identical or corresponding parts throughout the several views,FIG. 1 is a graph showing normalized transmission of power in aresonator. It is well known that the transmission of power through aresonator near a resonant frequency f_(q) can be expressed in the form:$\begin{matrix}{{T(\delta)} = \frac{T(0)}{1 + \left( {2Q_{L}\delta} \right)^{2}}} & {{Eq}.\quad 1}\end{matrix}$

where $\begin{matrix}{\delta = \frac{f - f_{q}}{f_{q}}} & {{Eq}.\quad 2}\end{matrix}$

is the normalized frequency. In these equations, f is the oscillationfrequency, f_(q) is the reference frequency (i.e., the resonantfrequency of the open resonator for an axial mode; e.g., a TEM_(00q)mode), Q_(L) is the loaded quality factor (i.e., the loaded Q) of theopen resonator, and T(δ) is the transmitted power. T(0) is thetransmitted power when the oscillation frequency equals the resonantfrequency (i.e., when f=f_(q)). As shown in FIG. 1, a plot of T(δ) as afunction of δ displays the well-known bell-shaped characteristic withits peak at δ=0.

One method of applying power to a resonator is via a voltage-controlledoscillator (VCO). In a VCO, the voltage applied to a control-inputterminal controls the oscillation frequency.

Even if the frequency, f_(s), of the VCO is slightly different fromf_(q), appreciable power may be transmitted through the open resonatorif f_(q) and f₅ are close. In practice, this requires that |f_(s)-f_(q)|is less than about f_(q)/Q_(L). From Equation 2, it follows that$\begin{matrix}{\delta_{s} = \frac{f_{s} - f_{q}}{f_{q}}} & {{Eq}.\quad 3}\end{matrix}$

Notably, δ₅ may be either positive or negative. Accordingly, it isassumed that the corresponding VCO control voltage is V_(S), which mayalso be either positive or negative depending on the device.

Using the difference between the oscillation frequency of the VCO andthe resonant frequency of a cavity resonator (either open or closed), itis possible to cause the frequency of oscillation of the VCO to lock onthe resonant frequency of the cavity as described below.

One such VCO is a varactor voltage controlled Gunn oscillator similar tothe Millitech Model Number WBV-28-220160RI, that operates in thefrequency range from about 34 GHz to about 36 GHz. As would beappreciated by one of ordinary skill in the art, the techniquesdescribed herein are also applicable to other kinds of voltagecontrolled oscillators that operate at very different frequencies(either higher or lower). For the Millitech device, V_(S) is negativeand the oscillation frequency increases as V_(S) becomes more negative.In the following description, behavior similar to the Millitech device'sis assumed.

Rather than using a fixed control voltage, it is possible to apply asmall amplitude time-varying voltage to the frequency controlling inputterminal of the VCO. Thus, the control voltage v_(c)(t) may be writtenin the form:

v _(C)(t)=V _(S) +v _(s)  Eq. 4a

where v_(s), the so-called “dither voltage,” is a function of time. Inone embodiment, the dither voltage is a symmetrical square wave. In analternate embodiment, the dither voltage is a sinusoidal signal.

Let

v _(s)(t)=v _(d) f(t)  Eq. 4b

where v_(d) is the peak amplitude of the dither voltage. Then, thefunction f(t) is bounded by −1 and 1. When using a symmetrical squarewave as a component of the dither voltage, the resulting normalizedfrequencies may be expressed in the form:

δ₊=δ₅ −a/2Q _(L)  Eq. 5a

when the square wave is positive, and in the form:

δ⁻=δ₅ +a/2Q _(L)  Eq. 5b

when the square wave is negative. That is, it is assumed that a smallsignal approximation is valid and that the change in δ is proportionalto the change in the control voltage. The positive parameter “a” isproportional to the square wave amplitude and has been introduced tofacilitate numerical calculations. Its value depends on the details ofthe relationship between the control voltage and the oscillationfrequency for the particular VCO, and it can be related to the amplitudeof the square wave “dither voltage.”

For the sake of discussion, δ_(s), is assumed to be positive (i.e., theoperating frequency for v₅=0 is greater than f_(q) so that the operatingpoint is to the right of the peak of the power transmission curve shownin FIG. 1).

From FIG. 1 and Equations 1, 5a, and 5b, it follows that the powertransmission through the resonator will be greater when the square waveis positive (δ=δ₊) than when the square wave is negative (δ=δ⁻). It isuseful to recognize that if δ_(s) were negative rather than positive,then the opposite would be true. A control voltage for the VCO can bederived from the difference between these two transmitted powers.Consider $\begin{matrix}{{\frac{{\Delta T}\left( \delta_{s} \right)}{T(0)} \equiv \frac{{T\left( {\delta_{s} - {{a/2}Q_{L}}} \right)} - {T\left( {\delta_{s} + {{a/2}Q_{L}}} \right)}}{T(0)}} = \frac{{T\left( \delta_{+} \right)} - {T\left( \delta_{-} \right)}}{T(0)}} & {{Eq}.\quad 6}\end{matrix}$

With the aid of Equation 1, it can be shown that: $\begin{matrix}{\frac{{\Delta T}\left( \delta_{s} \right)}{T(0)} = \frac{8Q_{L}\delta_{s}a}{\left\lbrack {1 + {\left( {2Q_{L}} \right)^{2}\left( {\delta_{s}^{2} + \left( {{a/2}Q_{L}} \right)^{2}} \right)}} \right\rbrack^{2} - \left( {4Q_{L}\delta_{s}a} \right)^{2}}} & {{Eq}.\quad 7}\end{matrix}$

and that the same expression is also valid if δ_(s) is negative.Equation 7 represents an asymmetric function of δ_(s) as shown in thegraph of FIG. 2.

The power transmitted through the resonator may be sampled and measuredusing a crystal diode detector, for which the voltage output is verynearly proportional to the power incident upon it. Therefore, a circuitthat can appropriately process the voltage output of a crystal diodethat responds to the power transmitted through the resonator can be usedto produce an output voltage that is essentially proportional toΔT(δ_(s)) as given by Equation 7.

The voltage produced by the control circuit will be positive when theoscillation frequency f_(s) is greater than the reference frequencyf_(q), 0 when f_(s)=f_(q), and negative when f_(s) is less than f_(q).This correction voltage v_(corr), which will typically be a slowlyvarying or quasi-DC voltage will be fed to the frequency controllingvoltage input terminal of the VCO in series with the DC voltage V_(s)and the dither voltage v_(s). Consequently, with the addition of thecorrection voltage v_(corr), Equation 4 becomes:

v _(C) =V _(S) +v _(s) +v _(corr)  Eq. 8

An exemplary range for the dither voltage amplitude (peak-to-peak) is 1to 50 mV. In general, the dither amplitude should be chosen such that itis less than approximately one half the detected voltage at the cavityresonance being tracked and, preferably as small as possible. Secondly,it should be noted that for robust locking over a range of RF power andpressure, the dither voltage amplitude and/or dither frequency may bevaried. Thirdly, the dither frequency should range from 1 kHz to 100kHz.

FIG. 3 is a block diagram of a stabilization system based on thetransmission properties discussed above. When the stabilization circuitis properly adjusted, the frequency of the VCO (with an internalisolator) 103 is locked to (i.e., is virtually the same as) the resonantfrequency of the open resonator 105. If the resonant frequency of theopen resonator 105 changes, the oscillation frequency of the VCO 103will change by an equivalent amount and the VCO 103 will continue toexcite the open resonator 105 at its changed resonant frequency.

As shown in FIG. 3, the VCO 103 is coupled to an open microwaveresonator 105. The microwave resonator 105 is used as a transmissioncavity and includes reflecting mirrors at each end. The resonator 105has separate input and output apertures, both of which may be on thesame mirror, or the input aperture may be on one mirror and the outputaperture on the other. The latter configuration may be preferable tolessen the likelihood that undesired off-axis modes will be excited ordetected.

Coupled to the output aperture of the resonator 105 is a detector 106(e.g., a crystal detector) for detecting the power transmitted throughthe resonator 105. The detector 106 and a dither voltage source 117 areconnected to a VCO bias signal generator 101 (shown in greater detail inFIG. 4). The outputs of (1) the VCO bias signal generator 101, (2) thedither voltage source, and (3) the DC bias source are fed to the adder102 that outputs the control voltage back to the VCO 103.

A block diagram of one embodiment of the VCO bias signal processor 101is shown in FIG. 4A. The dither (or clock) signal from dither voltagesource 117 is processed by high and low sample and hold circuits (124and 125, respectively) to provide high and low sample command signals,respectively, that act as the two phases of the dither signal. The highsample command signal permits the sample-and-hold circuit 124 to monitorthe output of amplifier 126 only when the dither (or clock) signal ispositive. Likewise, the low sample command signal permits thesample-and-hold circuit 125 to monitor the output of amplifier 126 onlywhen the dither (or clock) signal is negative. The relative timings ofthe command signals is shown in FIG. 4B. Differential amplifier 123provides the difference of the outputs of the sample-and-hold circuits124 and 125, and the integrator 122 limits the effects of transients toacceptable levels.

In addition, the integrator 122 facilitates locking the VCO 103 onto thedesired open resonator mode at start-up. For example, at start-up, theVCO DC bias voltage, V_(S), may be set to a value corresponding to anoscillator frequency, f_(s), slightly above the frequency of the desiredopen resonator mode, f_(q). The capacitor in the integrator 122 may thenbe charged to a voltage that when added algebraically to V_(S) by adder102 results in an oscillator frequency slightly less than f_(q). As theintegrator capacitor discharges, the frequency f_(s) will increase andapproach f_(q) and at some frequency during this approach thestabilization circuit will engage.

A block diagram of one embodiment of a plasma generator control systemaccording to the present invention is shown in FIG. 5. Other embodimentsare, of course, possible. Some of these alternate embodiments may notinclude such complete control of the circuit function by means of theDSP 109. Note that the determination of the control voltage for theplasma generator in the present invention is accomplished without theuse of a microwave discriminator, such as discriminators 110 and 110′ inFIGS. 4 and 5 of the co-pending application No. 60/144,880.

The digital signal processor (DSP) 109 shown in FIG. 5 responds to threeinputs: (1) the digital equivalent of the voltage at the frequencycontrolling input terminal of the VCO input to the DSP via A-to-D 108;(2) one of either the desired voltage at the frequency controlling inputterminal of the VCO, or the desired mean plasma density in the openresonator, or the desired frequency of the VCO; entered by means of akeyboard at the data input terminal or a potentiometer with the aid ofan A/D converter as shown in FIG. 5; and (3) the signal from a counterconnected to the detector at the output of the open resonator. The DSP109 provides output signals to control at least one of (1) the outputpower of the plasma generator; (2) the DC bias voltage for the VCO; (3)the amplitude of the dither voltage signal for the VCO; (4) the ditherfrequency; and (5) a display means. The output signal from the DSP tothe plasma generator 120 adjusts the power provided by the plasmagenerator to the plasma chamber 105 as required to achieve the desiredprocess parameters. FIG. 5 includes a directional coupler 104 and afrequency meter 118 between the VCO 103 and the plasma chamber 105 topermit an independent determination of the oscillator frequency wheneverit may seem prudent to do so. No discrete isolator is shown between theVCO and the plasma chamber in FIG. 5 because it is assumed that the VCOincludes an integral isolator. However, if the VCO does not include anintegral isolator, an isolator should be inserted in the circuit betweenthe VCO and the plasma chamber.

A change in the plasma generator control voltage causes the plasmadensity to change which in turn causes the resonant frequency of theopen resonator to change and, therefore, the output of the detector tochange. The detector signal causes the VCO bias signal processor outputto change and the voltage applied to the VCO frequency controlling inputto change. Consequently, the oscillation frequency of the VCO changes.The DSP compares the digital equivalent of the VCO control voltage tothe desired value based upon the input data and sends an appropriatecontrol signal to the plasma generator.

If the electromagnetic field in the open resonator collapses due to lossof control of the VCO frequency for any reason, the output of thedetector in the output line of the open resonator will drop to zero andthe counter 107 will record the event. If such an event takes place, thefrequency of the VCO is no longer controlled by the resonant frequencyof the open resonator, and “loss of lock” has occurred. Co-pendingapplication No. 60/144,880 considers in some detail the issue of “lossof lock.” Loss of frequency control can be used to instruct the DSP toinitiate an algorithm to reestablish frequency lock or to alert anequipment operator to loss of control and the possibility of equipmentmalfunction. It should be noted that a zero output from the detectorimplies that the VCO frequency is not locked to a resonant frequency ofthe open resonator.

The sampling period of the system is chosen to ensure that “loss oflock” will be recognized by the DSP. The DSP may then employ variousalgorithms to reestablish lock between the VCO frequency and theresonant frequency of the open resonator. A first such algorithmcalculates an expected plasma density based on one or more measuredparameters such as radio frequency (RF) power, gas pressure, gas flowrate, plasma chamber temperature, and plasma optical signature. The DSPthen searches for a discriminator zero (e.g., as shown in FIG. 2) withinprescribed frequency limits. A second such algorithm measures thefrequency differences between adjacent resonant modes to determine anapproximate plasma density and then conducts a search for adiscriminator zero in the neighborhood of the so-determined plasmadensity.

As depicted in FIG. 5, the VCO bias signal processor has been replacedwith a self-clocking VCO bias signal processor. In the embodimentillustrated in FIG. 6A, the VCO bias signal processor includes a VCObias signal processor 101 and a dither voltage source 117. In theembodiment illustrated in FIG. 6B, the VCO bias signal processorincludes a signal processor comprising a lock-in amplifier that includesa sinusoidal dither voltage signal. Lock-in amplifiers are especiallyuseful for processing weak signals in the presence of noise, which isadvantageous for the present application. The operation of lock-inamplifiers has been described elsewhere. (Lock-in amplifiers: principlesand applications, by M. L. Meade, Peter Peregrinus, Ltd., London, 1983.See also, Robert D. Moore, “Lock-in amplifiers for signals buried innoise,” Electronics, Vol. 35, pp. 40-43, Jun. 8, 1962.)

If the analysis of the resonant behavior of the open resonator isapproached from an optical perspective, rather than from a microwavecircuit perspective employed earlier in this disclosure, an equationthat is equivalent to Eq. 1 may be obtained. Let the resonant frequencyof an on-axis mode (e.g., a TEM_(00q) mode, which is the mode ofinterest here), be f_(q), where $\begin{matrix}{f_{q} \equiv {\frac{c}{2d}\left( {q + \frac{1}{2}} \right)}} & {{Eq}.\quad 9}\end{matrix}$

(Note: The usual notation for the resonant frequency of the TEM_(00q)mode is f_(00q), but because these are the only modes of interest here,the resonant frequency f_(00q) is represented by the simpler notationf_(q).)

The normalized power transmission through the open resonator T(f) may beexpressed by: $\begin{matrix}{{T(f)} = \frac{T\left( f_{q} \right)}{1 + {\left\lbrack \frac{2F}{\pi} \right\rbrack^{2}{\sin^{2}\left( \frac{2{\pi fd}}{c} \right)}}}} & {{Eq}.\quad 10}\end{matrix}$

where f is the frequency, c is the speed of light in vacuum, d is themirror separation, and F is the so-called finesse, which is defined as(c/2d) divided by the difference between the two frequencies nearestf_(q) for which T(f)=(1/2)T(f_(q)). Consider a Taylor series expansionof T(f) in the neighborhood of f_(s), where f_(s) is a frequency forwhich Eq. 10 is valid: $\begin{matrix}{{T(f)} \approx {{T\left( f_{s} \right)} + {\left( \frac{\partial T}{\partial f} \right)_{f = f_{s}}\left( {f - f_{s}} \right)} + {\frac{1}{2}\left( \frac{\partial^{2}T}{\partial f^{2}} \right)_{f = f_{s}}\left( {f - f_{s}} \right)^{2}} + \Lambda}} & {{Eq}.\quad 11}\end{matrix}$

In general the frequency change f−f_(s) is related to a correspondingchange in the control voltage; i.e., if f−f_(s) is not too large one maywrite $\begin{matrix}{{f - f_{s}} \approx {\left( \frac{\partial f}{\partial v_{C}} \right)\left( {v_{C} - V_{S}} \right)}} & {{Eq}.\quad 12}\end{matrix}$

where V_(S) is the DC control voltage for which the oscillator frequencyis f_(s), and the derivative is to be evaluated for v_(C)=V_(S).

Assume that the dither voltage is given by:

v _(d) =V _(d) cos(ω_(d) t)=v _(C) −V _(S)  Eq. 13

Then with the aid of Eqs. 11, 12, and 13, one may obtain an expressionfor the normalized average power transmitted through the cavity as afunction of time, where the average is taken over a time interval thatis long compared to the period of the microwave signal but shortcompared to the period of the dither signal. The result is:$\begin{matrix}{P_{AVG} \approx {{T\left( f_{s} \right)} + {\left( \frac{\partial T}{\partial f} \right)_{f = f_{s}}\left\lbrack {\left( \frac{\partial f}{\partial v_{C}} \right)_{v_{C} = V_{S}}V_{d}{\cos \left( {\omega_{d}t} \right)}} \right\rbrack} + {\frac{1}{2}{\left( \frac{\partial^{2}T}{\partial f^{2}} \right)_{f = f_{s}}\left\lbrack {\left( \frac{\partial f}{\partial v_{c}} \right)_{v_{C} = V_{S}}V_{d}{\cos \left( {\omega_{d}t} \right)}} \right\rbrack}^{2}} + K}} & {{Eq}.\quad 14}\end{matrix}$

The output voltage of a square law detector (e.g., a crystal detector)connected to the output port of the open resonator will be proportionalto P_(AVG). The second term in Eq. 14 is of special interest here, forit leads to the error signal for adjusting the frequency of the VCO. Inthis embodiment a lock-in amplifier is used to obtain the desired errorsignal.

The reference voltage v_(d) cos(ω_(d)t) of the lock-in amplifier alsoserves as the dither voltage. The lock-in amplifier includes anintegrator and produces a nominally time-independent output voltage,v_(out,) which may vary if the amplitude and phase of the input signalchange slowly with respect to the phase of the reference signal. For theapplication of interest here, the output voltage v_(out) isapproximately given by: $\begin{matrix}{v_{out} = {{K\left( \frac{\partial T}{\partial f} \right)}_{f = f_{s}}\left( \frac{\partial f}{\partial v_{C}} \right)_{v_{C} = V_{S}}V_{d}}} & {{Eq}.\quad 15}\end{matrix}$

where K is a constant of proportionality. Note that the algebraic signof v_(out) depends on whether f_(s) is greater than or less than f_(q),because the slope of T(f) is positive when f is less than f_(q) andnegative when f is greater than f_(q). (This algebraic sign change isequivalent to a 180 degree phase change.) The voltage v_(out) providesthe feedback voltage for controlling the frequency of the VCO 103through the adder 102. Depending on the voltage characteristic of theVCO 103, however, it may be necessary to insert an inverting amplifierbetween the lock-in amplifier and the adder to ensure that the algebraicsign of the error voltage at the frequency controlling input terminal ofthe VCO will cause the VCO frequency to move toward f_(q), rather thanaway from it.

The ability of a lock-in amplifier to detect weak signals in thepresence of noise is due to its use of an internal narrow-band amplifierand an internal integrator. However, the ability of a lock-in amplifierto amplify accurately a signal for which the amplitude and phase varywith time is limited by its so-called “settling time.” For use in theapplication considered here, the settling time must not be too great.Otherwise, the VCO may not be able to remain locked to the resonantfrequency of the open resonator as the plasma density changes,especially upon initiation of the plasma.

The digital signal processor 109 is programmed to control at least oneof (a) the output power of the plasma generator; (b) the DC bias voltagefor the VCO; (c) the reference (i.e., dither) voltage amplitude v_(d);(d) the reference signal frequency ω_(d); (e) the gain of the lock-inamplifier, (f) the settling time of the lock-in amplifier; and (e) adisplay means.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. It is thereforeto be understood that, within the scope of the appended claims, theinvention may be practiced otherwise than as specifically describedherein.

What is claimed is:
 1. A device for measuring electron density of a plasma in a plasma chamber comprising: an open microwave resonator coupled to said plasma chamber, and configured to be immersed in said plasma; a voltage controlled oscillator coupled to said open microwave resonator, said voltage controlled oscillator configured to be controlled by a feedback signal and configured to produce an excitation signal corresponding to at least one resonant mode in said open microwave resonator; a detector coupled to said open microwave resonator, and configured to measure said excitation signal, said feedback signal being generated based on an output of said detector; a dither voltage source for generating a dither voltage; and an adder for adding said dither control voltage source to said feedback signal and for applying an output of said adder to a voltage control input of said voltage controlled oscillator, wherein said output facilitates minimizing a difference between said excitation signal and said at least one resonant mode.
 2. The device of claim 1, wherein said open microwave resonator comprises a first mirror and a second mirror opposite said first mirror.
 3. The device of claim 2, wherein said voltage controlled oscillator is coupled to at least one of said first mirror and said second mirror.
 4. The device of claim 2, wherein said detector is coupled to at least one of said first mirror and said second mirror.
 5. The device of claim 1, wherein said dither voltage comprises at least one of a square wave and a sinusoidal signal.
 6. The device of claim 1, wherein the peak-to-peak amplitude of said dither voltage ranges from 1 to 50 mV.
 7. The device of claim 1, wherein the dither frequency of said dither voltage ranges from 1 to 100 kHz.
 8. The device of claim 1, further comprising a voltage control bias signal generator for generating said feedback signal from said detector.
 9. The device of claim 8, wherein the voltage control bias signal generator comprises an amplifier configured to receive said excitation signal from said detector and produce an amplified signal, a command signal generator configured to receive said dither voltage from said dither control voltage source, a high sample and hold device configured to receive said amplified signal from said amplifier and receive said dither voltage from said command signal generator, a low sample and hold device configured to receive said dither voltage from said command signal generator, a differential amplifier configured to receive a high sample and hold device output and a low sample and hold device output, and an integrator configured to receive a differential amplifier output and provide an input signal to said adder.
 10. The device of claim 1, wherein said open microwave resonator comprises a confocal resonator.
 11. A method of monitoring and controlling an electron density of a plasma generated by a plasma generator in a plasma chamber comprising: locking an excitation signal from a voltage controlled oscillator to at least one resonant mode in an open microwave resonator immersed in said plasma by controlling said voltage controlled oscillator using a feedback signal, wherein said feedback signal comprises a dither voltage and a correction signal based on an output of a detector coupled to the open microwave resonator; recording a first feedback signal; igniting a plasma in said plasma chamber after said first feedback signal is recorded; recording a second feedback signal after said plasma is ignited; and measuring an electron density of said plasma from a difference between said second feedback signal and said first feedback signal.
 12. The method of claim 11, wherein said open microwave resonator comprises a first mirror and a second mirror opposite said first mirror.
 13. The method of claim 12, wherein said voltage controlled oscillator is coupled to at least one of said first mirror and said second mirror.
 14. The method of claim 12, wherein said detector is coupled to at least one of said first mirror and said second mirror.
 15. The method of claim 11, wherein said dither voltage comprises at least one of a square wave and a sinusoidal signal.
 16. The method of claim 11, wherein the peak-to-peak amplitude of said dither voltage ranges from 1 to 50 mV.
 17. The method of claim 11, wherein the dither frequency of said dither voltage ranges from 1 to 100 kHz.
 18. The method of claim 11, further comprising generating said correction signal using a voltage control bias signal generator.
 19. The method of claim 18, wherein voltage control bias signal generator comprises an amplifier configured to receive said excitation signal from said detector and produce an amplified signal, a command signal generator configured to receive said dither voltage from a dither control voltage source, a high sample and hold device configured to receive said amplified signal from said amplifier and receive said dither voltage from said command signal generator, a low sample and hold device configured to receive said dither voltage from said command signal generator, a differential amplifier configured to receive a high sample and hold device output and a low sample and hold device output, and an integrator configured to receive a differential amplifier output and provide said correction signal to an adder.
 20. The method of claim 11, further comprising: specifying a target electron density; comparing said measured electron density with said target electron density; and adjusting said plasma generator in order to minimize a difference between said measured electron density and said target electron density.
 21. The method of claim 11, wherein said open microwave resonator comprises a confocal resonator.
 22. The method of claim 20, wherein said adjusting said plasma generator comprises adjusting a power coupled to said plasma by said plasma generator. 